The DOI matches its Crossref record; 59 of the 61 references it cites were found in a registry or an index.
Rational variability in children's causal inferences: the sampling hypothesis
Published version of a preprint
Abstract
We present a proposal-"The Sampling Hypothesis"-suggesting that the variability in young children's responses may be part of a rational strategy for inductive inference. In particular, we argue that young learners may be randomly sampling from the set of possible hypotheses that explain the observed data, producing different hypotheses with frequencies that reflect their subjective probability. We test the Sampling Hypothesis with four experiments on 4- and 5-year-olds. In these experiments, children saw a distribution of colored blocks and an event involving one of these blocks. In the first experiment, one block fell randomly and invisibly into a machine, and children made multiple guesses about the color of the block, either immediately or after a 1-week delay. The distribution of guesses was consistent with the distribution of block colors, and the dependence between guesses decreased as a function of the time between guesses. In Experiments 2 and 3 the probability of different colors was systematically varied by condition. Preschoolers' guesses tracked the probabilities of the colors, as should be the case if they are sampling from the set of possible explanatory hypotheses. Experiment 4 used a more complicated two-step process to randomly select a block and found that the distribution of children's guesses matched the probabilities resulting from this process rather than the overall frequency of different colors. This suggests that the children's probability matching reflects sophisticated probabilistic inferences and is not merely the result of a naïve tabulation of frequencies. Taken together the four experiments provide support for the Sampling Hypothesis, and the idea that there may be a rational explanation for the variability of children's responses in domains like causal inference.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“Bayesian samplers can match behavioral stochasticity in tasks such as perception, categorization, causal reasoning, and decision making [20,21,23,24].”
From Bayesian Brains without Probabilities (2016) · cites this paper as [24]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
A Handbook of Child Psychology
(1932) · Nature · doi:10.1038/130331a0 · Cited by 10,133
Monte Carlo Statistical Methods
Kim, H., Robert, C. P., & Casella, G. (2000) · Technometrics · doi:10.2307/1270959 · Cited by 5,596
(2004) · Choice Reviews Online · doi:10.5860/choice.42-1645 · Cited by 2,928
Piaget, J. (1976) · Piaget and His School · doi:10.1007/978-3-642-46323-5_2 · Cited by 2,167
Proceedings of the 29th Annual Conference of the Cognitive Science Society
Rohde, H., Kehler, A., & Elman, J. L. (2007) · Cognitive Science · Cited by 1,782
A Theory of Causal Learning in Children: Causal Maps and Bayes Nets.
Alison Gopnik, C., Glymour, C., Sobel, D. M., Schulz, L., Kushnir, T., & Danks, D. J. (2004) · Psychological Review · doi:10.1037/0033-295x.111.1.3 · Cited by 1,227
Emerging minds: the process of change in children's thinking
(1997) · Choice Reviews Online · doi:10.5860/choice.34-5984 · Cited by 1,132
GALTON, F. (1907) · Nature · doi:10.1038/075450a0 · Cited by 1,080
Word learning as Bayesian inference.
Xu, F., & Tenenbaum, J. B. (2007) · Psychological Review · doi:10.1037/0033-295x.114.2.245 · Cited by 1,061
Toward a statistical theory of learning.
Estes, W. K. (1950) · Psychological Review · doi:10.1037/h0058559 · Cited by 1,010
Proceedings of the 33rd Annual Conference of the Cognitive Science Society
Dziemianko, M., Keller, F., & Coco, M. I. (2011) · Cited by 737
Proceedings of the 31st Annual Conference of the Cognitive Science Society.
Tamariz, M. (2009) · Cited by 638
Kam, C. L. H., & Newport, E. L. (2005) · Language Learning and Development · doi:10.1080/15475441.2005.9684215 · Cited by 626
Young Children's Attribution of Action to Beliefs and Desires
Bartsch, K., & Wellman, H. M. (1989) · Child Development · doi:10.1111/j.1467-8624.1989.tb03526.x · Cited by 527
Vul (2014) · doi:10.1111/cogs.12101 · Cited by 502
Getting it right by getting it wrong: When learners change languages
Kam, C. L. H., & Newport, E. L. (2009) · Cognitive Psychology · doi:10.1016/j.cogpsych.2009.01.001 · Cited by 475
Alison Gopnik, C. (2012) · Science · doi:10.1126/science.1223416 · Cited by 469
A rational analysis of rule-based concept learning
Goodman, N. D., Tenenbaum, J. B., Feldman, J., & Griffiths, T. L. (2008) · Cognitive Science · doi:10.1080/03640210701802071 · Cited by 469
Proceedings of the 32nd Annual Conference of the Cognitive Science Society
Rohde, H., & Ettlinger, M. (2010) · Cited by 462
Young Children's Attribution of Action to Beliefs and Desires
Bartsch, K., & Wellman, H. M. (1989) · Child Development · doi:10.2307/1131035 · Cited by 416
An Economist’s Perspective on Probability Matching
Vulkan, N. (2000) · Journal of Economic Surveys · doi:10.1111/1467-6419.00106 · Cited by 413
Rational approximations to rational models: alternative algorithms for category learning
Sanborn, A. N., Griffiths, T. L., & Navarro, D. J. (2010) · Psychological Review · doi:10.1037/a0020511 · Cited by 369
Self-citationStudies in Mathematical Learning Theory.
Stone, M., Bush, R. R., & Estes, W. K. (1960) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2342659 · Cited by 355
Intuitive statistics by 8-month-old infants
Xu, F., & Garcia, V. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0704450105 · Cited by 353
Competitive foraging in mallards: “Ideal free’ ducks
Harper, D. G. C. (1982) · Animal Behaviour · doi:10.1016/s0003-3472(82)80071-7 · Cited by 341
Vul, E., & Pashler, H. (2008) · Psychological Science · doi:10.1111/j.1467-9280.2008.02136.x · Cited by 333
Sobel, D. M., Tenenbaum, J. B., & Alison Gopnik, C. (2004) · Cognitive Science · doi:10.1016/j.cogsci.2003.11.001 · Cited by 325
Pure Reasoning in 12-Month-Old Infants as Probabilistic Inference
Téglás, E., Vul, E., Girotto, V., Gonzalez, M., Tenenbaum, J., & Bonatti, L. (2011) · Science · doi:10.1126/science.1196404 · Cited by 296
Causal learning across domains.
Schulz, L., & Alison Gopnik, C. (2004) · Developmental Psychology · doi:10.1037/0012-1649.40.2.162 · Cited by 267
Development of proportional reasoning: Where young children go wrong.
Boyer, T. W., Levine, S. C., & Huttenlocher, J. (2008) · Developmental Psychology · doi:10.1037/a0013110 · Cited by 250
Intuitions of probabilities shape expectations about the future at 12 months and beyond
Téglás, E., Girotto, V., González, M., & Bonatti, L. L. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0700271104 · Cited by 227
The smart potential behind probability matching
Gaissmaier, W., & Schooler, L. J. (2008) · Cognition · doi:10.1016/j.cognition.2008.09.007 · Cited by 223
The Ecology of Foraging Behavior: Implications for Animal Learning and Memory
Kamil, A. C., & Roitblat, H. L. (1985) · Annual Review of Psychology · doi:10.1146/annurev.ps.36.020185.001041 · Cited by 211
Schulz, L., Bonawitz, E. B., & Griffiths, T. L. (2007) · Developmental Psychology · doi:10.1037/0012-1649.43.5.1124 · Cited by 193
The role of causality in judgment under uncertainty
Krynski, T. R., & Tenenbaum, J. B. (2007) · Journal of Experimental Psychology General · doi:10.1037/0096-3445.136.3.430 · Cited by 183
Kushnir, T., & Alison Gopnik, C. (2007) · Developmental Psychology · doi:10.1037/0012-1649.43.1.186 · Cited by 166
Developmental Differences in Rule Learning: A Microgenetic Analysis
Siegler, R. S., & Chen, Z. (1998) · Cognitive Psychology · doi:10.1006/cogp.1998.0686 · Cited by 162
The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities
Jeong, Y., Levine, S. C., & Huttenlocher, J. (2007) · Journal of Cognition and Development · doi:10.1080/15248370701202471 · Cited by 149
Occam's rattle: Children's use of simplicity and probability to constrain inference.
Bonawitz, E. B., & Lombrozo, T. (2011) · Developmental Psychology · doi:10.1037/a0026471 · Cited by 121
A more rational model of categorization
Sanborn, A. N., Griffiths, T. L., & Navarro, D. (2006) · Warwick Research Archive Portal (University of Warwick) · Cited by 109
Intuitive Theories of Mind: A Rational Approach to False Belief
Goodman, N. D., Baker, C. L., Mansinghka, V. K., Alison Gopnik, C., Wellman, H. M., Schulz, L., & Tenenbaum, J. B. (2005) · Cited by 94
Optimal Predictions in Everyday Cognition: The Wisdom of Individuals or Crowds?
Mozer, M. C., Pashler, H., & Homaei, H. (2008) · Cognitive Science · doi:10.1080/03640210802353016 · Cited by 90
Kushnir, T., Wellman, H. M., & Gelman, S. A. (2007) · Cognition · doi:10.1016/j.cognition.2007.10.004 · Cited by 81
Children’s understanding of posterior probability
Girotto, V., & González, M. (2007) · Cognition · doi:10.1016/j.cognition.2007.02.005 · Cited by 80
Modeling the effects of memory on human online sentence processing with particle filters
Lévy, R., Reali, F., & Griffiths, T. L. (2008) · neural information processing systems · Cited by 79
Comparison of predictions and estimates in a probability learning situation.
Neimark, E. D., & Shuford, E. H. (1959) · Journal of Experimental Psychology · doi:10.1037/h0043064 · Cited by 68
One and Done? Optimal Decisions from Very Few Samples
Goodman, N. D., Griffiths, T. L., Tenenbaum, J. B., & Vul, E. (2009) · eScholarship (California Digital Library) · Cited by 68
Integrating Physical Constraints in Statistical Inference by 11‐Month‐Old Infants
Denison, S., & Xu, F. (2010) · Cognitive Science · doi:10.1111/j.1551-6709.2010.01111.x · Cited by 66
Development of intuitive and numerical proportional reasoning
Ahl, V., Moore, C. F., & Dixon, J. A. (1992) · Cognitive Development · doi:10.1016/0885-2014(92)90006-d · Cited by 63
Bindra, D., Clarke, K. A., & Shultz, T. R. (1980) · Journal of Experimental Psychology General · doi:10.1037/0096-3445.109.4.422 · Cited by 56
Probability Learning and Sequence Learning
Estes, W. K. (2014) · doi:10.4324/9781315770406-13 · Cited by 43
Performing Bayesian Inference with Exemplar Models
Shi, L., Feldman, N. H., & Griffiths, T. L. (2008) · eScholarship (California Digital Library) · Cited by 33
EFFECTS OF AGE DIFFERENCES ON CHOICE BEHAVIOR1
Jones, M. H., & Liverant, S. (1960) · Child Development · doi:10.1111/j.1467-8624.1960.tb04982.x · Cited by 32
A Simple Sequential Algorithm for Approximating Bayesian Inference.
Bonawitz, E. B., Denison, S., Chen, A., Alison Gopnik, C., & Griffiths, T. L. (2011) · Cited by 27
Theory Acquisition as Stochastic Search
Ullman, T. D., Goodman, N. D., & Tenenbaum, J. B. (2010) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 23
Probabilistic Reasoning in Preschoolers: Random Sampling and Base Rate
Denison, S., Garcia, V. G., Konopczynski, K., & Xu, F. (2006) · eScholarship (California Digital Library) · Cited by 23
Is that your final answer? The effects of neutral queries on children’s choices
Gonzalez, A., Shafto, P., Bonawitz, E. B., & Alison Gopnik, C. (2012) · eScholarship (California Digital Library) · Cited by 13
Within-Subjects Procedural Variations in Two-Choice Probability Learning
Lehr, R., & Pavlik, W. B. (1970) · Psychological Reports · doi:10.2466/pr0.1970.26.2.651 · Cited by 7
Cited by
Sorted by citations
The Psychology and Neuroscience of Curiosity
Kidd, C., & Hayden, B. Y. (2015) · Neuron · doi:10.1016/j.neuron.2015.09.010 · Cited by 1,026
Computational rationality: A converging paradigm for intelligence in brains, minds, and machines
Gershman, S. J., Horvitz, E., & Tenenbaum, J. B. (2015) · Science · doi:10.1126/science.aac6076 · Cited by 722
Vul (2014) · doi:10.1111/cogs.12101 · Cited by 502
Confidence as Bayesian probability: From neural origins to behavior
Meyniel, F., Sigman, M., & Mainen, Z. F. (2015) · Neuron · doi:10.1016/j.neuron.2015.09.039 · Cited by 349
Alison Gopnik, C., O’Grady, S., Lucas, C. G., Griffiths, T. L., Wente, A., Bridgers, S., Aboody, R., Fung, H., & Dahl, R. E. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1700811114 · Cited by 355
Metadata
- Authors
- Denison, S., Bonawitz, E., Gopnik, A., & Griffiths, T. L.
- Affiliations
- University of Waterloo (CA)
- University of California, Berkeley (US)
- Corresponding author
- Denison, Stephanie
- Venue
- Cognition
- Publisher
- Elsevier BV
- Published
- 2013-02
- Volume
- 126
- Issue
- 2
- Pages
- 285-300
- ISSN
- 0010-0277 (ISSN-L)
- OpenAlex
- W2120212072
- Cited by
- 126
- References
- 61
- Citations by year
- 2026: 3 · 2025: 6 · 2024: 15 · 2023: 9 · 2022: 8 · 2021: 6 · 2020: 11 · 2019: 9 · 2018: 15 · 2017: 16 · 2016: 7 · 2015: 12 · 2014: 7 · 2013: 2
- FWCI
- 13.46
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Cognition (publishedVersion)
- PubMed (publishedVersion)
- Rutgers University Community Repository (Rutgers University) (submittedVersion)
- UWA Profiles and Research Repository (UWA) (submittedVersion)
- Topics
- Social Sciences › Psychology › Developmental and Educational Psychology › Child and Animal Learning Development
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Categorization, perception, and language
- Keywords
- causal inference, inductive reasoning, probability matching, preschool children, probabilistic reasoning, response variability, random sampling, subjective probability
- MeSH terms
- Child Development, Child, Preschool, Cognition, Concept Formation, Female, Humans, Male, Probability Learning, Problem Solving
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review